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To factor the quadratic expression \(2v^2 + 11v + 5\), we can follow these steps: 1. **Identify the coefficients**: - \(a = 2\) (coefficient of \(v^2\)) - \(b = 11\) (coefficient of \(v\)) - \(c = 5\) (constant term) 2. **Find two numbers that multiply to \(a \cdot c = 2 \cdot 5 = 10\) and add up to \(b = 11\)**: - The numbers 1 and 10 satisfy these conditions because \(1 \cdot 10 = 10\) and \(1 + 10 = 11\). 3. **Rewrite the middle term (11v) using the two numbers found**: \[ 2v^2 + 11v + 5 = 2v^2 + v + 10v + 5 \] 4. **Factor by grouping**: - Group the first two terms and the last two terms: \[ (2v^2 + v) + (10v + 5) \] - Factor out the greatest common factor (GCF) from each group: \[ v(2v + 1) + 5(2v + 1) \] 5. **Factor out the common binomial factor (2v + 1)**: \[ (v + 5)(2v + 1) \] So, the factored form of \(2v^2 + 11v + 5\) is: \[ (v + 5)(2v + 1) \]

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